Let A be an excellent local normal domain and {f(n)}(n=1)(infinity) a sequence of elements lying in successively higher powers of the maximal ideal, such that each hypersurface A/f(n)A satisfies R I We investigate the injectivity of the maps Cl(A) --> Cl((A/f(n)A)(1)), where (A/f(n)A)(1) represents the integral closure. The first result shows that no non-trivial divisor class can lie in every kernel. Secondly, when A is, in addition, an isolated singularity containing a field of characteristic zero, dim A greater than or equal to 4, and A has a small Cohen-Macaulay module, then we show that there is an integer N > 0 such that if f(n) epsilon m(N), then Cl(A) --> Cl((A/f(n)A)(1)) is injective. We substantiate these results with a general construction that provides a large collection of examples. (C) 2003 Elsevier B.V. All rights reserved. [References: 28]
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